Video: Computing Area of Parallelogram Using Matrices

Use determinants to calculate the area of the parallelogram with vertices (1, 1), (−4, 5), (−2, 8), and (3, 4).

02:27

Video Transcript

Use determinants to calculate the area of the parallelogram with vertices one, one; negative four, five; negative two, eight; and three, four.

So, what I’ve done here is that I’ve drawn a sketch. And this is a sketch of our parallelogram. So, how are we gonna use determinants to calculate the area? So, let’s consider how we’d actually work out the area of the parallelogram? Well, we’d multiply its width by its length.

So, in our diagram, we can represent these using vectors. So, our width is going to be two, three. So, it’s the vector two, three. And that’s because we’ve gone from negative four to negative two. So, that’s two in the positive direction. And then, we’ve also gone from five to eight. So, that’s up by three in the positive direction. And then, our vector for our length would be five, negative four. So, we’re gonna use these two vectors to determine the area of our parallelogram.

So, we’ve got the vectors two, three; five, negative four. And what we’re gonna do is we’re gonna put them together to form a two-by-two matrix where the columns are these two vectors. And it’s the determinant of this matrix that we’re gonna form that’s gonna find the area cause we can say that the area of the parallelogram is equal to the modulus of the determinant of the matrix two, five, three, negative four. So, now we’re gonna use the method we have to find the determinant of a two-by-two matrix.

And that is that if we have a matrix 𝑎, 𝑏, 𝑐, 𝑑, then the determinant of this matrix is equal to 𝑎𝑑 minus 𝑏𝑐. So therefore, we can say our area is going to be equal to the modulus, or absolute value, of two multiplied by negative four minus five multiplied by three, which is gonna be the absolute value, or modulus, of negative eight minus 15, which is gonna give us the result, which is the modulus, or absolute value, of negative 23.

Remembering, if we wanted to find the absolute value, then we disregard the negative result. Which makes sense in this situation because an area won’t be a negative value. So therefore, we can say that if we’ve got a parallelogram with vertices one, one; negative four, five; negative two, eight; and three, four, the area is going to be equal to 23.

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